Weibull-Gamma composite distribution: An alternative multipath/shadowing fading model

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1 Weibull-Gamma composite distribution: An alternative multipath/shadowing fading model Petros S. Bithas Institute for Space Applications and Remote Sensing, National Observatory of Athens, Metaxa & Vas. Pavlou Street, Palea Penteli, 1536 Athens, Greece ( Estimation of length: Units Column cm Title 80 chars 1 Affiliations 1 1 Abstract 530 chars 3 Text 6790 chars 35 References 8 8 Figures 17 Captions Equations Total: 81 June 13, 009

2 Weibull-Gamma composite distribution: An alternative multipath/shadowing fading model P.S. Bithas, N.C. Sagias and P.T. Mathiopoulos The Weibull-Gamma (WG) distribution, which is appropriate for modelling fading environments when multipath is superimposed on shadowing, is introduced and studied. For this composite distribution the probability density, cumulative distribution, characteristic functions and the moments are derived in closed form. Furthermore, the average bit error and outage probabilities of a receiver operating over WG fading channels are assessed and compared with the corresponding performances obtained using other composite distributions. Introduction: In many real life propagation links, multipath fading and shadowing occur simultaneously, yielding a composite fading environment [1]. Depending upon the nature of the radio propagation environment, several distributions have been proposed for modelling multipath fading, including Rayleigh, Nakagami and Weibull [1-], while shadowing is modelled by the lognormal distribution [1]. Based on these distributions various combinations have been suggested for obtaining the composite environment, including the widely accepted Rayleigh- and Nakagami-lognormal [1] and a relatively new one, the Weibull-lognormal (WL) [3]. A common characteristic of these lognormal-based composite distributions is their complicate mathematical form that limits their potential applications. An alternative approach is to employ the mathematical more convenient Gamma distribution for accurately modeling the shadowing effects [4]. Hence, using Gamma distribution simpler composite distributions have been proposed, such as the K [4] and the generalized-k [5]. Furthermore, in a recent work [6], the probability density function (PDF) of the 1

3 Weibull-Gamma (WG) fading model was presented in terms of the generalized hypergeometric function [6, Equation (5)]. However, to the best of the author s knowledge, a thorough analysis of the WG statistical characteristics is not available in the open technical literature and thus is the subject of the current work. Therefore, in this letter by deriving an alternative to [6, Equation (5)] representation for the PDF, the cumulative distribution function (CDF), the characteristic function (CF) and the moments of the WG composite distribution are presented. Capitalizing on these results the performance of a single receiver operating over such fading/shadowing channels is analyzed. Weibull-Gamma statistical characteristics: Let R representing the channel fading envelope following the Weibull distribution with PDF given by [1] ( ) ( β ) β/ β/ Γ 1+ / = exp Γ 1 +, 0 y β β 1 β/ fr x β x x y x (1) where β is distribution s shaping parameter, y is the average fading power y E R =, with E denoting expectation, and Γ( ) is the Gamma function [7, Equation (8.310/1)]. When multipath fading is superimposed on shadowing, y slow varies and hence it can be considered as a random variable following the Gamma distribution with PDF given by ( y Ω) α y ( ) 1 f exp / y y =, y 0 α Γ( α) Ω () where α > 0 is the shaping parameter and Ω= E y. Under these circumstances the PDF in (1) is conditioned on y, and in order to remove this conditioning (1) is averaged over () as

4 f ( x) = f ( x y) f ( y) dy. (3) R y y 0 Using the Meijer-G representation for the exponentials [8, Equation (11)], [7, Equation (9.31/)], [8, Equation (1)], and after some mathematical manipulations the PDF of the WG composite distribution can be expressed in closed form as f ( x) + β + 1 β 1 1/ + β / α βx Γ ( 1+ / β ),0 x Γ ( 1+ / β ) G = Γ ( α) 1 0, + Ω b Ω + ( π ) (4) where () G is the Meijer s G-function [7, Equation (9.301)], b ( ) Δ( α + β ) with ( ) 1,1 / = 1 Δ,1, + Δ x, y = y/ x,( y+ 1) / x,..., ( y + x 1)/ x, while and are positive integers properly chosen in order to satisfy / = β /. By varying α and β, the PDF of this generalized-weibull distribution can simultaneously describe several multipath and shadowing conditions. For instance for α, (4) approximates the Weibull distribution, for β = and using [7, Equation (9.34/3)], it simplifies to the K-distribution and hence approximates Rayleigh-lognormal distribution, while for β, α, (4) approaches the additive white Gaussian noise (AWGN) channel. Furthermore, by following a similar procedure as in [4], a relationship between the parameters α and β of WG distribution and the mean, μ, and standard deviation, σ, of the WL can be derived as μ = ln ( ) ( α), σ '( α) ln 10 Ω +Ψ = Ψ (5) ln 10 ( ) where Ψ() and Ψ' () are the psi function and its derivative respectively [7, Equation (8.360/1)]. Substituting (4) in the definition of the nth order moments of, ( ) n ( ) Χ μ n E, making a change of variables and using [7, Equation (7.811/4)], a closed-form expression for ( n) μ can be obtained as 3

5 μ ( n) ( 1 / β ) n/ n/ β + 3/ a+ n/ 3/ + β Γ + β + n = Γ bi. + + Ω (6) Γ i= 1 1 ( π) ( α) Furthermore, using the product theorem for the Gamma functions [7, Equation (8.335)] and after some mathematical manipulations, (6) simplifies to μ ( n) ( n ) ( n ) Γ( α ) ( ) Γ 1 + / β Γ α + / Γ 1+ / β = Ω n/. (7) x Substituting (4) in the definition of the CDF of, ( ) ( ) change of variables and using [8, Equation (6)] yields F x f x dx, making a 0 β / 1/ α ( β+ 3)/ β β Γ ( 1+ / β ) x +,1 x Γ ( 1+ / β ) 1 1/ F ( x) = G ( )( ) ( ). + / 1 α π 1, Ω Γ Ω b+, 1/ (8) The CF of is defined as Φ () s E exp( js) Χ, where j = 1. Starting from this definition, using (4) and [8, Equations (11) and (1)], Φ ( s) can be obtained as Χ β / β 1/ 1/ α+ β/ 1 Γ ( 1+ / β ) β +, Γ ( 1+ / β ) Δ(,1 β ) Φ () s = G. (3+ 3)/ ( ) ( ) ( ), js π α + b Ω Γ Ω( js) + (9) Single receiver: Let us consider a receiver operating over WG composite fading channels. The equivalent baseband received signal can be expressed as r = sh+ n, where s is the transmitted complex symbol with energy Es = E s, n is the complex AWGN with single sided power spectral density N 0 and h is the channel complex gain, i.e., = h. The instantaneous signal-to-noise ratio (SNR) per received symbol is given by γ = / N, while using (7), i.e., setting n =, the Es 0 corresponding average SNR can be obtained as γ = =Ω. E Es / N0 αes / N0 Using the last two expressions in (8), the CDF of γ can be easily obtained as 4

6 F γ ( γ ) β / 1/ α ( β+ 3)/ β/ Γ ( 1+ / β ) β γ +,1 [ αγ ( 1+ / β) γ ] 1 1/ = G. + γ / α 1, + + 1, 1/ ( α)( π b ) ( γ) + Γ (10) For the receiver under consideration, the outage probability, P out, defined as the probability that γ falls below a certain specified threshold, γ th, can be easily obtained, using (10), as P out = F γ ( γ th ). Moreover, by making a change of variables in (4) and following a similar procedure as for deriving (9), the moments generating function (MGF) of γ, defined as M () s E exp( sγ ) γ, can be obtained as β / 1/ α 1 Γ ( 1+ / β ) β / +, [ αγ ( 1+ / β) ] Δ(,1 β / ) Mγ () s = G.(11) + 3 γs / α, + ( α)( π b Γ ) ( γ s) + By using (11) and following the MGF-based approach, direct calculation of the average bit error probability (ABEP) for non-coherent binary frequency shift keying and differential binary phase shift keying (DBPSK) is possible [1]. For example, the ABEP of DBPSK is given by Pbe = 0.5 (1) [1]. M γ Numerical results: In Fig.1, the P out is plotted as a function of the normalized outage threshold, γ / γ, for several values of α and β. It is depicted that as γ / th th γ and/or α, β increase the outage performance improves. It should be noted that the curves for β = correspond to K fading channels, while it can be easily observed that the gap among the curves increases as α increases. In Fig., the ABEP of a single receiver operating over the equivalent WG and WL composite fading channels is plotted as a function of γ. The ABEP for WG fading channels is determined using (11), while for the WL numerical integration techniques have been employed. Moreover, in order to obtain equivalent channel conditions, (5) is used for selecting appropriate values for α and σ. In Fig., it is evident that the ABEP performances for these fading 5

7 channel models are very close, and this close agreement further improves as α increase. It is noted that similar ABEP performance results have been obtained for other modulation schemes, e.g., M-ary PSK. Conclusions: In this letter the most important statistical characteristics, such as PDF, CDF, CF and moments of the WG composite distribution have been derived. Using these results, the performance analysis of a receiver operating over WG fading channels was investigated in terms of the ABEP and the P out. Numerically evaluated results have confirmed that the WG distribution can be used as an alternative to the common, but much more complex, employed Rayleigh- and Weibull-lognormal distributions as a reliable ABEP predictor in multipath/shadowing fading channels. References [1] SIMON, M., and ALOUINI, M.: Digital communication over fading channels (New York, John Wiley, 000, nd edn.). [] SAGIAS, N., and KARAGIANNIDIS, G., Gaussian class multivariate Weibull distribution: Theory and applications in fading channels, IEEE Trans. Inf. Theory, 51 (10), pp , Oct [3] ISMAIL, M., and MATALGAH, M., Outage probability in multiple access systems with Weibull-faded lognormal-shadowed communication links Proc. IEEE VTC 05, Dallas, Tex, USA, Sep [4] ABDI, A., and KAVEH, M.: K distribution: An appropriate substitute for Rayleigh-lognormal distribution in fading-shadowing wireless channels, Electron. Lett., 1998, 34, (9), pp

8 [5] BITHAS, P., SAGIAS, N., MATHIOPOULOS, P., KARAGIANNIDIS G., and RONTOGIANNIS, A.: On the performance analysis of digital communications over Generalized-K fading channels, IEEE Commun. Lett., 4 (10), pp , May 006. [6] NADARAJAH, S., and KOTZ, S.: A class of generalized models for shadowed fading channels, Wireless Pers. Commun., 007, 43, (), pp [7] GRADSHTEYN, I.S., and RYZHIK, I.M.: Table of integrals, series, and products (New York, Academic, 000, 6th edn.). [8] ADAMCHIK, V., and MARICHEV, O. The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system, Proc. Intern. Conf. on Symbolic and Algebraic Computation, Tokyo, Japan, 1990, pp

9 Figure 1 1 outage probability (P out ) α = 1 α = 5 β = β = 3 β = normalized outage threshold (db) Fig. 1 Outage probability versus the normalized outage threshold for several values of α and β. 8

10 Figure average bit error probability (ABEP) β =, α = 3.5 σ =.5 β = 3, α = 4 σ =.3 β = 1.5, α = σ = 3.5 β = 4, α = 5 σ = 10-4 Weibull-Gamma Weibull-Lognormal average input SNR (db) Fig. The ABEP of WG and WL fading channels versus the average input SNR for several values of a, β and σ. 9

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